The proposed research aims at: a) Improving the current knowledge about the linkage map, by testing for new markers, blood samples from 2,000 families extensively types for other genetic markers under previous NIH grants. Large numbers of possible linkages will then be available for testing with comparatively little new laboratory work and expense. b) Improving the understanding of the genetics of inherited diseases with unclear mode of transmission by using as markers alleles at well-defined loci. This will be done by formal mathematical analysis of genetic patterns taking into consideration all information about the diverse biases incurred in the selection of the families and will focus on the possibility of uncovering residual genetic heterogeneity in those diseases. The data obtained in our own studies of Type I diabetes mellitus (JDM) and those of others in the literature will be used as the empirical control of the mathematical models to be developed. We shall also study the mathematical problems associated with the definition of linkage in the presence of population association between the disease and specific alleles at linked loci and will explore further the problem of the verification of the existence of such associations. We have already proposed and partially tested a new statistic to replace the "relative risk" (RR) estimate of Woolf and Haldane which we call the "haplotype relative risk" (HRR) and which is not affected by heterogeneity between samples of patients and controls. As a result of the advances expected from this work, we should be able to define the linkage relationships of several "new" polymorphic traits with the 25 "old" ones already defined in our sample and among themselves. We also expect to use the new mathematical tools to be developed for the study of diseases of complex genetics, such as JDM, with a view to increasing the precision of genetic counselling and, more importantly, to suggesting approaches for the study of the mechanisms involved.